String theory was originally formulated from a perturbative description (using quantum mechanics (QM) and replacing points by strings and evaluating path integral). Still, although QM has an upgrade to a field formalism -- QFT (that helps to solve the problem with causality and negative energy particles), I don't really see why people want to look for a field formalism of string theory. What problem does String Theory share with QM? What are the advantages of a string field theory? Why the underlying degrees of freedom for string theory might be fields?
The different string vibration modes represent particles. There are infinitely many different excitations of increasing mass and experiments usually have a limited energy scale E, so we don't worry about particles with m>E. One can then only consider e.g. massless modes and then build an effective theory describing their interactions. One can do this by working out their string interactions and then write down a Lagrangian for a field theory describing the given set of particles and interactions. Why exactly field theory? Relativistic theories of elementary particles are under some assumptions always quantum field theories (c.f. Weinberg). Why is this useful? We know how to handle QFT!